Specifically, I remember a song with “Mama look-a…” and “Poppa” in it. Not a long song, apparently sung by a group.
Also, “Freight Train,” parodied in MAD as:
Space ship, space ship, go so fast!
space ship, space ship, shoot right past!
Earth is no more place to stop
Since the H-bomb make it pop!
For this one, I never knew the artist’s name.
The parodied song is
Freight Train - by - The Chas McDevitt skiffle group(featuring Nancy Whiskey)
I think I have this somewhere on a 78 and I believe the group was still playing until very recently.
The first one is “Mama Look-a Boo-boo”, by Harry Belafonte. Very big in 1957.
It was also heard in the film Howard Stern’s Private Parts.
In the US, the song Freight Train by Rusty Draper hit the charts at the same time as the McDivitt group in England.
More than likely, Draper’s version in the US would have been the source of the parody.
Thanx to Neidhart and others. 
Harry Belafonte, no less!
Belafonte’s lyrics are funny! I’d forgotten the lyrics after all those years 